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COR. 2. Hence an strat glit line drawn throughth centre, an terminate by opposite hyperbolas, is bisected in the centre. Coll. 3. I CD, EF meet an hyperbola, O Fig. lo. iis opposite hyperbola again in the potnis , - -
equat, as also BF, E. Coll. 4. And sincerit has been proved that Fig. Io. the quare of the semidiameter AO, OM is n. a.
equat to the rectangle EBF that is, o EΝ; there rem is to AO, as A t EN and consequently BN is greater than 25. 5. Elem.)twice O that is, than AM; that is, any transverse diameter is tessalian any ther str ight line parallela it, an terminate in opposite hyperbolaS.COR. 5. I in a traight linei terminated by the hyperbolas, there heuaken theloinis E, F such, that each of the rectangles BEN, BF be equat to the square of the semidiameter A
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whicli is parallel o B the potnis E, F are in the asymptoteS. PROP. XVI. THEOR.
I Domin pol ni in an hyperbola tothe Symptote any Wo traight lines e drawn to hicli ther two traight line drawn to the asymptote sto any the poliat in the Same, o opposite hyperbola, are parallel the rectangle contained by the forme straight lines is qua to that contained by the lalter.
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Dra through the potnis A, B to the asymptotes the stra ight lines AH, BL parallelto the seconi a Xis, and the propo Sition may bedemonstra ted in the fame ord With the pre-Ceding. . COR. I. Hence, Piso tW potnis in an hy-Perbola, o opposite hyperbolas, omne or both of the asymptotes, wo traight lines bydra wn parallel to the ther asymptote, o to both of thema the rectangle containe di ea cli paralleland the abscissa a et ween i and the centre
Let A, B be the potnis; through them dra AC and BE, o BF parallel to the Symptotex the reC tangi contained by the parallel AC, and the abscissa Coae tween AC and thecentre, is equat to the rectangle BEO. BFO.
In generat, the paris cut T in an indesinite traight Iine, undistimated Domin giveta poliat by parallel draWnsrom an curve line, and formin With it a givei angie, are Callem AbacisSae, or absciSSas and the Parallel are called Ordinate to that curVC.
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the centre, and with in the an-
gle contained by the asymptotes, me et the hyperbola.
Fig. 12. Le AB, AC e the asymptotes, and ADthe hal of the transverse axis, and let AE bean strat glit line drawn through the Centre, and passing ith in the angi BAC; his stra ight line A meet the hyperbola. For fAE meetsno the hyperbola, through D dra BD parallel to the secon axis, and meetin the
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fore meet the transverse axis at right an gles,
and so illiu the hyperbola II 3: let it cutit in H on the fame fide of AD, illi the poliat
F and et i meet themther asymptot in K: since, there re the poliaticis ithout the hyperbola, GH is greater than F, that is, than BD and H is greater than DC: therei rethe rectangle GH is greater than the rectan-gle BDC, that is, than the quare of BD: ut, by the 13th of this book, the rectangle HKis equa to the square of BD whicli is absurd. There re AE necessarii meet the hyperbola. Coll. I seo the centre a strat glit line Aie Fig. Io.drawn with in the angle contained by the asymp- - totes and the quare of this traight in beequa to the rectangle EBF containe by the segments of an strat glit line parallel o A, whicli are intercepte between the oin B, where that paralle meet the hyperbola, and the potnis E, F heresi meet the asymptotes ;
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the poliat Acis in ne of the hyperbolas. For, according to the proposition the traight lineo necessari ly meet the hyperbola; is, there- fore, it meeis no the hyperbola in A, it mustmeet it in sonae ther pol ni P and then thesqua re of OP Willi equa to the l. cor. 15. 3. rectangle EBF that is to the quare of
A Whicli is absurdo Therei ore the poliat is in the hyperbola. PROP. XVIII. Prop. I 3. B. 2. Apoll. γI with in the angi contained by the a Symptotes, any traight linebe raWn parallel to ei ther of the asymptote M it me et the hyperbola in ne poliat Ialy, and passe SV ith in the hyperbola.